May I recommend two improvements?

• If you're ok with there being only a single chain — I'd even think it was preferable — then

      @matching{@names} = shuffle(@names); # Match 'em up
      
      # No one has themselves
      foreach (@names) {
          next GENERATE if $_ eq $matching{$_};
      }
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  simplifies to

      %matching = ( map { $_, $_ } @names )[ 1..@names*2-1, 0 ];
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• Another common limit is that B doesn't give to A if A gave to B the previous year. That can be achieved by changing

  push @digest, md5_hex($_, '=>', $matching{$_});
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  to

  push @digest, md5_hex(join '<=>', sort $_, $matching{$_});
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someone actually bothers to try to crack it

That's *very* easy to do. Say you have "N" participants.

• To find out from whom someone specific receives requires at most N hashes, half that on average.
• To find out to whom someone specific gives requires at most N hashes, half that on average.
• To find all relations requires at most (N*(N-1)-1)/2 hashes, half that on average.

It's even easier than attacking passwords of length 2. Hashing the pairings is merely security theatre. You might as well MIME::Base64 them.